Structural properties of Dirichlet series with harmonic coefficients

An infinite family of functional equations in the complex plane is obtained for Dirichlet series involving harmonic numbers. Trigonometric series whose coefficients are linear forms with rational coefficients in hyperharmonic numbers up to any order are evaluated via Bernoulli polynomials, Gauss sum...

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Veröffentlicht in:The Ramanujan journal Jg. 45; H. 2; S. 569 - 600
Hauptverfasser: Alkan, Emre, Göral, Haydar
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer US 01.02.2018
Springer Nature B.V
Schlagworte:
Coefficients
Combinatorics
Dirichlet problem
Field Theory and Polynomials
Fourier Analysis
Functional equations
Functions (mathematics)
Functions of a Complex Variable
Mathematical analysis
Mathematics
Mathematics and Statistics
Number Theory
Real numbers
Representations
Polylogarithm
11M41
Harmonic number
30B50
Functional equation
33B10
30D05
Dirichlet series
Gauss sum
Catalan’s constant
ISSN:1382-4090, 1572-9303
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Zusammenfassung:An infinite family of functional equations in the complex plane is obtained for Dirichlet series involving harmonic numbers. Trigonometric series whose coefficients are linear forms with rational coefficients in hyperharmonic numbers up to any order are evaluated via Bernoulli polynomials, Gauss sums, and special values of L -functions subject to the parity obstruction. This in turn leads to new representations of Catalan’s constant, odd values of the Riemann zeta function, and polylogarithmic quantities. Consequently, a dichotomy result is deduced on the transcendentality of Catalan’s constant and a series with hyperharmonic terms. Moreover, making use of integrals of smooth functions, we establish Diophantine-type approximations of real numbers by values of an infinite family of Dirichlet series built from representations of harmonic numbers.
Bibliographie:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
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ISSN:1382-4090
1572-9303
DOI:10.1007/s11139-017-9906-5